The numerical solving of linear and partial differential equations is a subject that Cheney and Kincaid’s Numerical Mathematics and Computing discusses thoroughly.  We have yet to cover it in class, but I feel it is one of the most important fields of numerical computation.  Finite element analysis is a significant and widely used application of this topic. Finite element analysis uses the finite element method, which is a strategy for solving partial differential equations.  It is used to simulate real world situations that cannot be solved analytically because there are usually too many factors involved and the system is not well approximated by a regular geometry.  Since most of the physical world can be described by differential equations of some sort, finite element analysis is a technique that has applications over many disciplines in science and engineering.

Applications:

            Finite element analysis is commonly used in civil and mechanical engineering in stress analysis.  It is also known for its common use in problems involving transfer processes such as in heat transfer and fluid dynamics.  A process with changing boundary conditions or the study of diffusion in rigid bodies may also cause difficulties in finding a closed-form analytic solution, in which case finite element analysis may be very useful in understanding the system.  One more specific application is in the field of Microelectromechanical systems or MEMS.  The last reference attached below is an in depth discussion of finite element analysis as it applies to this subject.

Basic Explanation:

            What finite element analysis essentially does is breaks a complicated system up into small, manageable pieces.  These manageable pieces are often a set of nodes for which certain physical characteristics can be calculated such as the displacement, speed, temperature, etc.  The nodes together form a mesh which contains information about how the system parts interact, such as applied forces and the properties of the materials.  Particular nodes to pay attention to are those that are perhaps near boundaries or have a special interaction with the system as a whole.  For instance, in a stress analysis application of finite element analysis, the nodes involved would have information about how they are loaded, and an engineer or scientist using this analysis would pay particular attention to known sources of fracture.  From here, properties can be numerically considered or certain quantities may be minimized or maximized to find ways of optimizing the system based on anticipated uses of the system.  Finite element analysis is especially good at showing potential issues that may otherwise be difficult to predict.  Knowing these issues may make the difference between a successful product or experiment and an unsuccessful one; in many cases, it can make the difference between human safety and danger, which can easily be realized upon consideration of the engineering of a vehicle, for example, that must be as collision safe as possible.  The key to finding these issues and properties is the use of the finite element method which can be learned about in section 15.3 of Cheney and Kincaid’s Numerical Mathematics and Computing, but lacking that, several of the references below give a further explanation of this method.

Finite element analysis is an incredible useful modeling system that allows numerical computation to better affect human concerns.  It has applications in a wide range of vital areas of engineering and experimental science, and it makes the world safer by improving many products that we depend on regularly.  Finite element analysis directly uses numerical methods to give important results that would otherwise be impossible to obtain.

References:

http://mathworld.wolfram.com/FiniteElementMethod.html

http://en.wikipedia.org/wiki/Finite_element_method

http://audilab.bmed.mcgill.ca/AudiLab/teach/fem/fem.html

http://en.wikipedia.org/wiki/Finite_element_analysis

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/num/widas/history.html

http://parts.jpl.nasa.gov/docs/JPL%20PUB%2099-1G.pdf